Identity-based simplification with cubes and squares Evaluate [137^2 + 137×133 + 133^2] / [137^3 − 133^3] using algebraic factorization.

Introduction / Context:
This expression is designed to test recognition of classic algebraic identities. The numerator matches a^2 + ab + b^2 and the denominator matches a^3 − b^3 for a = 137 and b = 133. Recognizing these patterns allows immediate reduction without heavy arithmetic.

Given Data / Assumptions:

• a = 137, b = 133.
• Numerator: a^2 + ab + b^2.
• Denominator: a^3 − b^3.

Concept / Approach:
Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). When a^3 − b^3 is divided by a^2 + ab + b^2, the result is simply a − b. Therefore, (a^2 + ab + b^2) / (a^3 − b^3) = 1 / (a − b).

Step-by-Step Solution:
Numerator N = a^2 + ab + b^2.Denominator D = a^3 − b^3 = (a − b)(a^2 + ab + b^2) = (a − b)·N.Therefore, N / D = N / [(a − b)·N] = 1 / (a − b).Compute a − b = 137 − 133 = 4.Value = 1/4.

Verification / Alternative check:
Direct cancellation is valid because (a^2 + ab + b^2) is nonzero for distinct positive a and b. Hence, the simplified ratio is exact.

Why Other Options Are Wrong:
4 and 270 invert the correct ratio; 1/270 confuses the difference with a product; 1/8 is an arbitrary fraction not supported by the identity.

Common Pitfalls:
Attempting long numerical computations instead of using identities; misremembering a^3 − b^3 factorization; cancelling the wrong factor.

Final Answer:
1/4